A result of great theoretical and experimental interest, Jarzynski equality predicts a free energy change $\Delta F$ of a system at inverse temperature $\beta$ from an ensemble average of non-equilibrium exponential work, i.e., $\langle e^{-\beta W}\rangle =e^{-\beta\Delta F}$. The number of experimental work values needed to reach a given accuracy of $\Delta F$ is determined by the variance of $e^{-\beta W}$, denoted ${\rm var}(e^{-\beta W})$. We discover in this work that ${\rm var}(e^{-\beta W})$ in both harmonic and an-harmonic Hamiltonian systems can systematically diverge in non-adiabatic work protocols, even when the adiabatic protocols do not suffer from such divergence. This divergence may be regarded as a type of dynamically induced phase transition in work fluctuations. For a quantum harmonic oscillator with time-dependent trapping frequency as a working example, any non-adiabatic work protocol is found to yield a diverging ${\rm var}(e^{-\beta W})$ at sufficiently low temperatures, markedly different from the classical behavior. The divergence of ${\rm var}(e^{-\beta W})$ indicates the too-far-from-equilibrium nature of a non-adiabatic work protocol and makes it compulsory to apply designed control fields to suppress the quantum work fluctuations in order to test Jarzynski equality.